Since the first lunar missions in the 1960s, the moon has been the object of interest of both scientific research and potential commercial development. During the 1980s, several lunar missions were launched by national space agencies. Interest in the moon is increasing with the advent of the multi-national space station making it possible to stage lunar missions from low earth orbit. However, continued interest in the moon and the feasibility of a lunar base will depend, in part, on the ability to schedule frequent and economical lunar missions.
A typical lunar mission comprises the following steps. Initially, a spacecraft is launched from earth or low earth orbit with sufficient impulse per unit mass, or change in velocity, to place the spacecraft into an earth-to-moon orbit. Generally, this orbit is a substantially elliptic earth-relative orbit having an apogee selected to nearly match the radius of the moon's earth-relative orbit. As the spacecraft approaches the moon, a change in velocity is provided to transfer the spacecraft from the earth-to-moon orbit to a moon-relative orbit. An additional change in velocity may then be provided to transfer-the spacecraft from the moon-relative orbit to the moon's surface if a moon landing is planned.
FIG. 1 is an illustration of orbits of a conventional lunar mission in a non-rotating coordinate system wherein the X-axis 10 and Y-axis 12 are in a plane defined by the moon's earth-relative orbit 36. The Z-axis 18 to the X and Y axes. In the conventional lunar mission, the spacecraft is launched 27 from earth 16 or low earth orbit 20, not necessarily circular, and has sufficient velocity to place the spacecraft into an earth-to-moon orbit 22, which crosses the moon's sphere of influence orbit 30.
Near the moon 14, a change in velocity reduces the spacecraft's moon-relative energy and transfers the spacecraft into a moon-relative orbit 24, which is not necessarily circular. An additional change in velocity transfers the spacecraft from the moon-relative orbit 24 to the moon by way of the moon landing trajectory 25. When an earth-return is desired, a change in velocity is sufficient to place the spacecraft into a moon-to-earth orbit 26 either directly from the moon's surface or indirectly by multiple impulses during the descent. Finally, near the earth 16, a change in velocity reduces the spacecraft's earth-relative energy and returns the spacecraft to low earth orbit 20 or to earth 16, via the earth-return trajectory 27
Herein the states the spacecraft are defined as positions and velocities of the spacecraft. To estimate the state, various methods have been used. The typical lunar mission is designed within a two-body problem framework, based on the Hohmann transfer as described above. That approach leads to trajectories that can be completed in a few days. However, the Hohmann transfer is suboptimal with respect to fuel consumption. Those trajectories usually tend to tolerate noise, and uncertainties. Estimation methods based on a Kalman filter (such as an extended Kalman filter (EKF), Batch-Kalman filter) have been used to estimate the state of the spacecraft in the presence of noise and uncertainty. Monte-Carlo based particle filters have been used to perform very accurate estimation for some missions, and various design phase studies.
The conventional methods for estimating trajectories have some important drawbacks. For example, the methods that use the Kalman filter techniques are inaccurate due to sensitive dynamics of three-body problem. The methods that use the Monte-Carlo filter are extremely complex computationally.
Recently, advances have been made to obtain greater understanding of a planar circular restricted three-body problem (PCR3BP) that considers the gravity of the moon. There is great interest in utilizing trajectories that can use less fuel than the Hohmann-transfer trajectories. However, the three-body problem is chaotic and highly sensitive to initial conditions, and various types of noise. The resulting trajectories take less fuel than conventional design, but are more sensitive to perturbations. This implies that the spacecraft on one of these trajectories can go off course due to very small perturbations in very short time.
Accordingly, there is a need for a method that can efficiently and accurately estimate spacecraft trajectories from an earth orbit to a moon orbit, and can be used on-board of a space craft for real time estimation of a state of the spacecraft while minimizing fuel consumption.